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Some Omega results for Dirichlet $L$ -functions

Motivated by prior work, this paper employs the resonance method to establish new Omega results for Dirichlet $L$ -functions, thereby extending existing knowledge in the field.

Original authors: Qiyu Yang, Shengbo Zhao

Published 2026-05-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Qiyu Yang, Shengbo Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are exploring a vast, invisible forest made entirely of numbers. In this forest, there are special trees called Dirichlet L-functions. Mathematicians have been studying these trees for a long time because they hold the secrets to how prime numbers (the building blocks of all numbers) are distributed.

Sometimes, these trees grow very tall, and sometimes they stay short. The big question this paper asks is: "How tall can these trees get?"

The authors, Qiuyu Yang and Shengbo Zhao, are trying to prove that under certain conditions, these mathematical trees can grow surprisingly high—much higher than we might expect just by looking at the average.

Here is a simple breakdown of their journey and findings:

1. The Goal: Finding the "Tallest" Tree

In the world of these functions, there are many different "versions" of the tree, depending on a specific setting called a "modulus" (think of this as the type of soil or the specific garden plot). The authors are looking at gardens where the soil is a prime number (a very specific, indivisible type of soil).

They want to find the single version of the tree in that garden that reaches the highest point. They aren't just guessing; they want to prove a mathematical "floor" for how high it must go.

2. The Tool: The "Resonance Method"

To find the tallest tree, the authors use a clever trick called the Resonance Method.

The Analogy: Imagine you are in a room full of different musical instruments (the different versions of the L-functions). You want to find the one that plays the loudest note. Instead of listening to each one individually, you play a specific "resonance" chord. This chord is designed to vibrate in perfect harmony with the loudest instrument, making it stand out even more, while silencing the others.
In the Paper: They construct a special mathematical "chord" (called a resonator). When they apply this to their forest of trees, it amplifies the signal of the tallest tree, allowing them to measure its height with great precision.

3. The Main Discovery: Proving the Height

The authors prove that for any large enough prime number (any large enough garden), there is always at least one character (one specific version of the tree) that grows to a height defined by a specific formula involving logarithms.

What this means: They have established a new, stronger "minimum height" guarantee. Before this, we knew the trees could get tall, but this paper says, "No, they can get this tall, and here is the proof."
The "Omega" Result: In math, an "Omega result" is like saying, "This thing will definitely reach at least this high, no matter how you try to stop it."

4. Two Different Scenarios

The paper looks at the problem in two ways:

Scenario A (The General Case): They prove the height result without needing to assume any unproven theories. They use a "sharper" way of counting zeros (dead spots in the forest) to get a better estimate than previous researchers.
Scenario B (The "GRH" Case): They also look at what happens if we assume a famous, unproven theory called the Generalized Riemann Hypothesis (GRH) is true. If this theory holds, the "resonance" works even better, and they can prove the trees can reach an even more specific, optimized height.

5. Why It Matters (According to the Paper)

The authors mention that this work improves upon a paper written by one of the authors' colleagues (Daodao Yang) in 2024. By using a more precise way of counting the "dead spots" (zeros) in the forest and choosing their parameters more carefully, they have managed to tighten the constants in their formulas.

In simple terms: They didn't just find a new tree; they built a better ruler to measure the tallest trees, proving they are taller than previously thought possible.

Summary

This paper is a mathematical proof that uses a "resonance" technique to show that Dirichlet L-functions can reach specific, very large values. It refines previous work by using better counting methods, ensuring that for any large prime number, there is always a version of these functions that grows to a guaranteed, impressive height.

Technical Summary: Some Omega Results for Dirichlet L-Functions

Problem Statement
The paper addresses the problem of establishing lower bounds (Omega results) for the large values of Dirichlet $L$ -functions, specifically $L(\sigma, \chi)$ where $\sigma \in (1/2, 1)$ and $\chi$ is a non-principal Dirichlet character modulo a large prime $q$ . The study focuses on the magnitude of $\log |L(\sigma, \chi)|$ and the real part of rotated logarithms $\Re(e^{-i\theta} \log L(\sigma, \chi))$ . This problem is central to analytic number theory due to its connections with character sums, quadratic non-residues, and class numbers. Previous work by Montgomery (for the Riemann zeta function) and Aistleitner et al. (for Dirichlet $L$ -functions) established lower bounds of the order $(\log q)^{1-\sigma} / (\log \log q)^\sigma$ . The authors aim to refine the constants in these bounds and extend the results to rotated logarithms.

Methodology
The authors employ the resonance method, a technique previously utilized by the first author in 2024. The core of the methodology involves the following steps:

Resonator Construction: A resonator function $R(\chi)$ is constructed as a Dirichlet polynomial defined by a completely multiplicative function $r(n)$ . The function is supported on integers composed of primes up to $X = A(\log q)(\log \log q)$ , with coefficients $r(p) = 1 - (p/X)^\sigma$ .
Truncation and Exceptional Sets: The logarithm of the $L$ -function, $\log L(\sigma, \chi)$ , is approximated by a truncated sum $T_\chi(\sigma, Y) = \sum_{p \le Y} \chi(p)p^{-\sigma}$ . This approximation holds for all characters except a set of "exceptional" characters $E_q$ . The size of this exceptional set is controlled using zero-density estimates for $L(s, \chi)$ (Lemma 2).
Moment Estimation: The authors analyze the weighted sums of the resonator over the group of characters $G_q$ . Specifically, they compare the total moment $Q_1 = \sum |R(\chi)|^2$ and the weighted moment $Q_2 = \sum \Re(e^{-i\theta} T_\chi) |R(\chi)|^2$ against the sums restricted to non-exceptional characters ( $S_1$ and $S_2$ ).
Optimization: By choosing the parameter $A$ appropriately (specifically $A < \frac{1-\vartheta(\sigma-\epsilon)}{\sigma(1+\lambda(\sigma))}$ ), they ensure that the contribution of the exceptional characters is negligible compared to the total sum. This allows the maximum value over non-principal characters to be bounded from below by the ratio $Q_2/Q_1$ .

Key Contributions and Results

Theorem 1 (Unconditional Result): For any fixed $\sigma \in (1/2, 1)$ and sufficiently large prime $q$ , the authors prove:
$\max_{\chi \neq \chi_0} \Re(e^{-i\theta} \log L(\sigma, \chi)) \ge C_{\sigma, \theta} \frac{(\log q)^{1-\sigma}}{(\log \log q)^\sigma}$
for any $\theta \in [0, \pi/2] \cup [3\pi/2, 2\pi]$ . The constant $C_{\sigma, \theta}$ is explicitly bounded and depends on $\sigma$ and $\theta$ .
- Corollary 1: By setting $\theta = 0$ , this yields a lower bound for $\max \log |L(\sigma, \chi)|$ . The authors claim this improves the constant $C(\sigma)$ found in their previous work [13] by utilizing a sharper zero-density estimate (Lemma 2) and more careful parameter selection.
Theorem 2 (Conditional Result under GRH): Assuming the Generalized Riemann Hypothesis (GRH) for $L(s, \chi)$ , the authors establish a similar lower bound with a different constant $D_{\sigma, \theta}$ . The GRH assumption simplifies the zero-density estimates, leading to a slightly different form for the constant, specifically involving a factor of $(2\sigma)^{-(1-\sigma)}$ .
Extension to Derivatives: In the concluding remarks, the authors note that their method, combined with a sharper zero-density estimate, can also be applied to the logarithmic derivative $-L'/L(\sigma, \chi)$ . They recover a result consistent with Daodao Yang's recent work [14], establishing a lower bound of order $(\log q)^{1-\sigma}(\log \log q)^{1-\sigma}$ .

Significance and Claims
The paper claims to extend previous results on the large values of Dirichlet $L$ -functions by:

Refining Constants: Improving the explicit constants in the lower bounds for $\log |L(\sigma, \chi)|$ compared to the 2024 work of the first author [13].
Generalizing the Argument: Extending the results to the real part of rotated logarithms $\Re(e^{-i\theta} \log L(\sigma, \chi))$ , rather than just the modulus.
Clarifying Modulus Conditions: The authors emphasize that their results hold for sufficiently large prime moduli $q$ , addressing a point raised in Daodao Yang's work [14] regarding the necessity of $q$ being prime for certain estimates to hold uniformly.

The authors do not claim to resolve the conjectured upper bounds (Montgomery's conjecture) but rather provide improved lower bounds that are closer to the conjectured optimal order, up to the multiplicative constant. The work is presented as a refinement of existing resonance method techniques applied to the specific context of Dirichlet $L$ -functions modulo primes.